294 Proceedings of Boyal Society of Edinburgh, [june 20 , 
z 
X 
3-67049 
8-63412 
- *40276 
12-30461 
13-57025 
4-93613 
+ -30012 
25-87486 
18-50533 
4-93508 
- -24700 
44-38019 
23-44022 
4-93489 
+ -21836 
67-82041 
- -19647 
Here the second differences of z are seen to he in excess of J7J- 2 , 
and to tend towards it. These maximum points are below the 
middles of their respective arcs by the distances 
•86131 
•86517 
•86605 
•86638 
•86654 
which evidently approximate to some definite limit. The exact 
determination of this asymptote would he a matter of great diffi- 
culty. 
In passing from side to side of its axis the chain must change the 
direction of its curvature, the concavity being in general toward the 
axis ; hut the points of reflexure are not necessarily at the crossings. 
At these points the second derivative must he zero ; now the very 
genesis of the curve is contained in the equation 
— x = lz x -f z 2z x or 2 z x — - lz , 
wherefore for the points of reflexure D, G, K, the ordinate x and its 
derivative lz x must he equal to each other with opposite signs. The 
x dx 
suhtangent of the curve is given by the formula or ~ x—- 
CtX 
in Leibnitz’ notation; wherefore the tangents applied at these points 
must meet the axis at the distance of unit (that is the length of the 
corresponding pendulum) above the points d , g , h, h of the figure, 
as also is the case for the tangent applied at the lowest point A. 
